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Arc Length Calculator

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How to Calculate the Area of a Sector and the Length of an Arc

The area of a sector and the length of an arc can be calculated using this handy arc length calculator.

In the coming paragraphs, we explain the formula for arc length and give a guide with detailed instructions on how to calculate the arc length. The equation for sector area is also given.

The radius and central angle \(\theta\) are the key components for determining the length of the arc. It is known that the arc length is equal to the circumference for an angle equal to 360 degrees or 2π radians.  From this information, it can be determined that \(\frac{L}{\theta }\) = \(\frac{C}{2\pi}\).  Since it is known that \(C = 2\pi r\), by substitution we obtain \(\frac{L}{\theta} = \frac{2\pi r}{2\pi}\).  Through algebraic cancellation, \(\frac{L}{\theta} = r\).  To eliminate the denominator of the fraction on the left side of the equation, multiply by \(\theta\). This yields the formula for the arc length L.

L = r*\(\theta\)  Therefore, the arc length is radius multiplied by the central angle (in radians).

Now, for the area of a sector of a circle.

It is known that the area of a circle is \(\pi r^{2}\).  From the use of proportions, it can be said that

\(\frac{A}{\theta} = \frac{\pi r^{2}}{2\pi}\), which after cancellations becomes \(\frac{A}{\theta} = \frac{r^{2}}{2}\).  Solving the equation for A, we know the area of a sector is given by \(A = r^{2}\times \frac{\theta}{2}\)

Here’s an example finding the length of an arc and area of a sector:

Suppose the radius of the circle is 10 cm. (You can also use the diameter and enter either information in the calculator)

Now choose a central angle.  Let’s say it is equal to 60 degrees, or \(\frac{\pi}{3}\).

Using the formula we obtained for arc length, we can solve for L : \(L = r\times \theta = 10 \times \frac{\pi}{3} = 10.47\text{ cm}\).

Calculate the area of a sector: \(A = r^{2}\times\frac{\theta}{2} = 10^{2}\times\frac{\frac{\pi}{3}}{2} = 52.36\text{ cm}^{2}\).

With our calculator you can enter the angle in degrees, radians, or gradians.  The radius or diameter can be entered from millimeters to kilometers or from inches to miles.